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Recursively defined polynomials, read by row.
5

%I #7 Jun 24 2020 03:10:00

%S 1,0,1,0,1,1,0,2,1,1,1,0,4,2,7,4,6,2,1,1,0,8,4,30,25,72,68,116,102,

%T 118,84,67,33,16,4,1,1,0,16,8,124,114,641,776,2495,3372,7637,10444,

%U 18561,24212,35684,42828,53707,57884,62257,59056,54261,44356,34326,23548

%N Recursively defined polynomials, read by row.

%C lim(N->infinity)p_N(x)/prod(n = 0, N-1)p_n(x) = f(x), with f(x) from A109087.

%H Robert Israel, <a href="/A109086/b109086.txt">Table of n, a(n) for n = 1..9957</a>

%F p_0(x) = x, p_(n+1)(x) = p_n(x)^2 + prod(i = 0, n-1)p_i(x)^(n-i), n >= 0. p_n(x) = sum(i = 0, 2^n)t(n, i)*x^i, n >= 0. a(2^(n+1)+n-i)=t(n, i), n>=0, 0<=i<=2^n.

%e p_0(x) = x, p_1(x) = x^2 + 1, p_2(x) = x^4 + 2*x^2 + x + 1,

%e p_3(x) = x^8 + 4*x^6 + 2*x^5 + 7*x^4 + 4*x^3 + 6*x^2 + 2*x + 1

%e p_4(x) = x^16 + 8*x^14 + 4*x^13 + 30*x^12 + 25*x^11 + 72*x^10 + 68*x^9 + 116*x^8 + 102*x^7 + 118*x^6 + 84*x^5 + 67*x^4 + 33*x^3 + 16*x^2 + 4*x + 1

%p p[0]:= x:

%p for n from 1 to 7 do

%p p[n]:= expand(p[n-1]^2 + mul(p[i]^(n-1-i),i=0..n-2))

%p od:

%p for n from 0 to 7 do

%p seq(coeff(p[n],x,j),j=degree(p[n]) .. 0,-1)

%p od; # _Robert Israel_, Jun 23 2020

%o (PARI) N=5;p=x;q=1;r=1;for(n=0,N,print(Vec(p));q*=r;r*=p;p=p^2+q)

%Y Cf. A109088, A109089, A109090.

%K easy,nonn,tabf,look

%O 1,8

%A Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jun 19 2005